Bonnet et al. [J. ACM, 69 (2022), 3] introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdös--Rényi random graph $G(n,p)$ with $1/n\leq p\leq 1/2$ is larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for any positive $\varepsilon$. Last, we calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$.


  1. twin-width
  2. random graphs
  3. extremal
  4. Paley graph

MSC codes

  1. 05C35
  2. 68Q87

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Information & Authors


Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 2352 - 2366
ISSN (online): 1095-7146


Submitted: 14 October 2021
Accepted: 18 July 2022
Published online: 26 September 2022


  1. twin-width
  2. random graphs
  3. extremal
  4. Paley graph

MSC codes

  1. 05C35
  2. 68Q87



Funding Information

Institute for Basic Science https://doi.org/10.13039/501100010446 : IBS-R029-C1

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